JEE MAIN Matrices Previous Year Question Paper and Solution with PDF

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JEE MAIN Previous Year Questions (PYQs)

JEE MAIN Matrices PYQ

Qus : 1 JEE MAIN PYQ

If the system of equations $x+\big(\sqrt{2}\sin\alpha\big)y+\big(\sqrt{2}\cos\alpha\big)z=0$ $x+(\cos\alpha)y+(\sin\alpha)z=0$ $x+(\sin\alpha)y-(\cos\alpha)z=0$ has a non-trivial solution, then $\alpha\in\left(0,\frac{\pi}{2}\right)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Morning Shift) PYQ

Solution

Qus : 2 JEE MAIN PYQ

If $P = \begin{bmatrix} 1 & 0 \\ \tfrac{1}{2} & 1 \end{bmatrix}$, then $P^{50}$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Evening Shift) PYQ

Solution

Qus : 3 JEE MAIN PYQ

Let $A=\begin{bmatrix}1\\1\\1\end{bmatrix}$ and $B=\begin{bmatrix} 9^{2} & -10^{2} & 11^{2}\\ 12^{2} & 13^{2} & -14^{2}\\ -15^{2} & 16^{2} & 17^{2} \end{bmatrix}$, then the value of $A'BA$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 July Evening Shift) PYQ

Solution

Qus : 4 JEE MAIN PYQ

If $A$ is a $3\times 3$ non-singular matrix such that $AA' = A'A$ and $B = A^{-1}A'$, then $BB'$ equals :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2014 (Offline) PYQ

Solution

Qus : 5 JEE MAIN PYQ

Let the sum of the focal distances of the point $P(4,3)$ on the hyperbola $H:\ \dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ be $8\sqrt{\dfrac{5}{3}}$. If for $H$, the length of the latus rectum is $l$ and the product of the focal distances of the point $P$ is $m$, then $9l^{2}+6m$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Evening Shift) PYQ

Solution

Qus : 6 JEE MAIN PYQ

If $\alpha,\beta \ne 0$, and $f(n) = \alpha^{n} + \beta^{n}$ and $\begin{vmatrix} 3 & 1 + f(1) & 1 + f(2) \\ 1+f(1) & 1 + f(2) & 1 + f(3) \\ 1+f(2) & 1 + f(3) & 1 + f(4) \end{vmatrix} = K(1-\alpha)^{2}(1-\beta)^{2}(\alpha-\beta)^{2}$, then $K$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2014 (Offline) PYQ

Solution

Qus : 7 JEE MAIN PYQ

For the system of linear equations $\alpha x+y+z=1,\quad x+\alpha y+z=1,\quad x+y+\alpha z=\beta$, which one of the following statements is **NOT** correct?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (1 February Evening Shift) PYQ

Solution

Qus : 8 JEE MAIN PYQ

The values of $\lambda$ and $\mu$ for which the system of linear equations \[ \begin{aligned} x + y + z &= 2,\\ x + 2y + 3z &= 5,\\ x + 3y + \lambda z &= \mu \end{aligned} \] has infinitely many solutions are, respectively:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Morning) PYQ

Solution

Qus : 9 JEE MAIN PYQ

Let m and M be respectively the minimum and maximum values of \[ \left| \begin{array}{ccc} \cos^{2}x & 1+\sin^{2}x & \sin 2x\\ 1+\cos^{2}x & \sin^{2}x & \sin 2x\\ \cos^{2}x & \sin^{2}x & 1+\sin 2x \end{array} \right|. \] Then the ordered pair (m, M) is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Morning) PYQ

Solution

Let $m$ and $M$ be respectively the minimum and maximum values of \[ \left|\begin{matrix} \cos^2 x & 1+\sin^2 x & \sin 2x\\ 1+\cos^2 x & \sin^2 x & \sin 2x\\ \cos^2 x & \sin^2 x & 1+\sin 2x \end{matrix}\right|. \] Then the ordered pair $(m, M)$ is equal to : Solution: Apply $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_1$ \[ \Delta = \left|\begin{matrix} \cos^2 x & 1+\sin^2 x & \sin 2x\\ 1 & -1 & 0\\ 0 & -1 & 1 \end{matrix}\right| \] Expanding along the first row, \[ \Delta = \cos^2 x(-1) - (1+\sin^2 x)(1) - \sin 2x(1) \] \[ \Delta = -(\cos^2 x + \sin^2 x + 1 + \sin 2x) \] \[ \Delta = -2 - \sin 2x \] Since $\sin 2x \in [-1,1]$, \[ \Delta \in [-3, -1] \] Hence, $m = -3$, $M = -1$ Therefore, the ordered pair is $\boxed{(-3, -1)}$.

Qus : 10 JEE MAIN PYQ

A value of $\theta \in \left( {0,{\pi \over 3}} \right)$, for which
$\left| {\matrix{ {1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } & {4\cos 6\theta } \cr {{{\cos }^2}\theta } & {1 + {{\sin }^2}\theta } & {4\cos 6\theta } \cr {{{\cos }^2}\theta } & {{{\sin }^2}\theta } & {1 + 4\cos 6\theta } \cr } } \right| = 0$, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Evening Shift) PYQ

Solution

Qus : 11 JEE MAIN PYQ

Let $A = \left[ {\matrix{ 1 & 2 \cr { - 1} & 4 \cr } } \right]$. If A^$-$1 = $\alpha$I + $\beta$A, $\alpha$, $\beta$ $\in$ R, I is a 2 $\times$ 2 identity matrix then 4($\alpha$ $-$ $\beta$) is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 July Morning Shift) PYQ

Solution

Qus : 12 JEE MAIN PYQ

The system of equations $\begin{cases} x+y+z=6,\\ x+2y+5z=9,\\ x+5y+\lambda z=\mu \end{cases}$ has no solution if:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (23 January Evening Shift) PYQ

Solution

Qus : 13 JEE MAIN PYQ

If $A=\dfrac12\begin{bmatrix}1 & \sqrt{3}\\ -\sqrt{3} & 1\end{bmatrix}$, then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (1 February Evening Shift) PYQ

Solution

Qus : 14 JEE MAIN PYQ

Let $A=\left[a_{i j}\right]$ be a $3 \times 3$ matrix such that $A\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], A\left[\begin{array}{l}4 \\ 1 \\ 3\end{array}\right]=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]$ and $A\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$, then $a_{23}$ equals :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (23 January Evening Shift) PYQ

Solution

Qus : 15 JEE MAIN PYQ

$ \text{Let the minimum value } v_{0} \text{ of } v=\lvert z\rvert^{2}+\lvert z-3\rvert^{2}+\lvert z-6i\rvert^{2},\ z\in\mathbb{C} \text{ be attained at } z=z_{0}. \text{ Then } \lvert 2z_{0}^{2}-\overline{z_{0}}^{\,3}+3\rvert^{2}+v_{0}^{2} \text{ is equal to:} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 July Morning Shift) PYQ

Solution

Qus : 16 JEE MAIN PYQ

Let $A=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$ and $B=I+\operatorname{adj}(A)+(\operatorname{adj} A)^2+\ldots+(\operatorname{adj} A)^{10}$. Then, the sum of all the elements of the matrix

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Evening Shift) PYQ

Solution

Qus : 17 JEE MAIN PYQ

If the system of equations $x + y + a z = b$ $2x + 5y + 2z = 6$ $x + 2y + 3z = 3$ has infinitely many solutions, then $2a + 3b$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (6 April Morning Shift) PYQ

Solution

Qus : 18 JEE MAIN PYQ

Let $A$ be a matrix such that $A \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$ is a scalar matrix and $|3A| = 108$. Then $A^2$ equals :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Morning Shift) PYQ

Solution

Qus : 19 JEE MAIN PYQ

Let $\theta = {\pi \over 5}$ and $A = \left[ {\matrix{ {\cos \theta } & {\sin \theta } \cr { - \sin \theta } & {\cos \theta } \cr } } \right]$. If B = A + A⁴, then det (B) :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Evening) PYQ

Solution

Qus : 20 JEE MAIN PYQ

Let $A=\begin{pmatrix} 0 & 2q & r\\ p & q & -r\\ p & -q & r \end{pmatrix}$. If $AA^{T}=I_{3}$, then $|p|$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Morning Shift) PYQ

Solution

Qus : 21 JEE MAIN PYQ

Let $A = [a_{ij}]_{2\times 2}$, where $a_{ij}\ne 0$ for all $i,j$ and $A^{2}=I$. Let $a$ be the sum of all diagonal elements of $A$ and $b=\lvert A\rvert$ (i.e., $b=\det A$). Then $3a^{2}+4b^{2}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (6 April Morning Shift) PYQ

Solution

Qus : 22 JEE MAIN PYQ

If the system of linear equations

$2x+2y+3z=a$

$3x-y+5z=b$

$x-3y+2z=c$

where $a,b,c$ are non-zero real numbers, has more than one solution, then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Morning Shift) PYQ

Solution

Qus : 23 JEE MAIN PYQ

The number of values of $k$, for which the system of equations : $\matrix{ {\left( {k + 1} \right)x + 8y = 4k} \cr {kx + \left( {k + 3} \right)y = 3k - 1} \cr } $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2013 (Offline) PYQ

Solution

Qus : 24 JEE MAIN PYQ

Let $P$ be a square matrix such that $P^{2}=I-P$. For $\alpha,\beta,\gamma,\delta\in\mathbb{N}$, if $P^{\alpha}+P^{\beta}=\gamma I-29P$ and $P^{\alpha}-P^{\beta}=\delta I-13P$, then $\alpha+\beta+\gamma-\delta$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (6 April Evening Shift) PYQ

Solution

Qus : 25 JEE MAIN PYQ

If the system of linear equations

$x + ay + z = 3$

$x + 2y + 2z = 6$

$x + 5y + 3z = b$

has no solution, then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Evening Shift) PYQ

Solution

Qus : 26 JEE MAIN PYQ

Consider the matrix $f(x)=\begin{bmatrix} \cos x & -\sin x & 0\\ \sin x & \cos x & 0\\ 0 & 0 & 1 \end{bmatrix}$. Given below are two statements: Statement I : $f(-x)$ is the inverse of the matrix $f(x)$. Statement II : $f(x)f(y)=f(x+y)$. In the light of the above statements, choose the correct answer:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Morning Shift) PYQ

Solution

Qus : 27 JEE MAIN PYQ

If $A = \left( {\matrix{ {{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr {{{ - 2} \over {\sqrt 5 }}} & {{1 \over {\sqrt 5 }}} \cr } } \right)$, $B = \left( {\matrix{ 1 & 0 \cr i & 1 \cr } } \right)$, $i = \sqrt { - 1} $, and Q = A^TBA, then the inverse of the matrix A Q²⁰²¹ A^T is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 August Morning Shift) PYQ

Solution

Qus : 28 JEE MAIN PYQ

For the system of equations \[ \begin{cases} x+y+z=6,\\ x+2y+\alpha z=10,\\ x+3y+5z=\beta, \end{cases} \] which one of the following is **NOT** true?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (6 April Evening Shift) PYQ

Solution

Qus : 29 JEE MAIN PYQ

Suppose $A$ is any $3\times 3$ non-singular matrix and $(A-3I)(A-5I)=0$ where $I=I_{3}$ and $O=O_{3}$. If $\alpha A+\beta A^{-1}=4I$, then $\alpha+\beta$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Evening Shift) PYQ

Solution

Qus : 30 JEE MAIN PYQ

Let $A = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 1 \cr 1 & 0 & 0 \cr } } \right)$. Then A²⁰²⁵ $-$ A²⁰²⁰ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 August Evening Shift) PYQ

Solution

Qus : 31 JEE MAIN PYQ

If the system of equations $2x - y + z = 4$, $5x + \lambda y + 3z = 12$, $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Morning Shift) PYQ

Solution

Qus : 32 JEE MAIN PYQ

The values of a and b, for which the system of equations 2x + 3y + 6z = 8, x + 2y + az = 5, 3x + 5y + 9z = b, has no solution, are :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Morning Shift) PYQ

Solution

Qus : 33 JEE MAIN PYQ

Let $A = [a_{ij}]$ be a square matrix of order $2$ with entries either $0$ or $1$. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $\mathrm{P}(E)$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Evening Shift) PYQ

Solution

Qus : 34 JEE MAIN PYQ

Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu$, and a system of linear equations, x + y + z = 5, x + 2y + 3z = $\mu$ ,x + 3y + $\lambda$z = 1, is constructed. If p is the probability that the system has a unique solution and q is the probability that the system has no solution, then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 August Evening Shift) PYQ

Solution

Qus : 35 JEE MAIN PYQ

$ \text{Let } I(x)=\int \frac{x+1}{x,(1+x e^{x})^{2}},dx,; x>0.\ \text{If } \lim_{x\to\infty} I(x)=0,\ \text{then } I(1) \text{ is equal to:} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (8 April Morning Shift) PYQ

Solution

Qus : 36 JEE MAIN PYQ

If the system of equations $x+2y-3z=2$, $2x+\lambda y+5z=5$, $14x+3y+\mu z=33$ has infinitely many solutions, then $\lambda+\mu$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Evening Shift) PYQ

Solution

Qus : 37 JEE MAIN PYQ

Let the system of equations

x + 5y - z = 1

4x + 3y - 3z = 7

24x + y + λz = μ

λ, μ ∈ ℝ, have infinitely many solutions. Then the number of the solutions of this system,

if x, y, z are integers and satisfy 7 ≤ x + y + z ≤ 77, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (7 April Evening Shift) PYQ

Solution

Qus : 38 JEE MAIN PYQ

$ \text{Let } A \text{ and } B \text{ be any two } 3\times 3 \text{ symmetric and skew-symmetric matrices respectively. Then which of the following is NOT true?} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 July Evening Shift) PYQ

Solution

Qus : 39 JEE MAIN PYQ

Let A be a 3 $\times$ 3 matrix with det(A) = 4. Let R_i denote the i^th row of A. If a matrix B is obtained by performing the operation R₂ $ \to $ 2R₂ + 5R₃ on 2A, then det(B) is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Evening Shift) PYQ

Solution

Qus : 40 JEE MAIN PYQ

Let $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $Q=P A P^{T}$. If $P^{T} Q^{2007} P=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$, then $2 a+b-3 c-4 d$ equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (8 April Morning Shift) PYQ

Solution

Qus : 41 JEE MAIN PYQ

If the matrix $A = \left( {\matrix{ 0 & 2 \cr K & { - 1} \cr } } \right)$ satisfies $A({A^3} + 3I) = 2I$, then the value of K is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 August Morning Shift) PYQ

Solution

Qus : 42 JEE MAIN PYQ

Let $A$ be a $3 \times 3$ real matrix such that $A\begin{pmatrix}1 \\ 1 \\ 0\end{pmatrix} = \begin{pmatrix}1 \\ 1 \\ 0\end{pmatrix}, \quad A\begin{pmatrix}1 \\ 0 \\ 1\end{pmatrix} = \begin{pmatrix}-1 \\ 0 \\ 1\end{pmatrix}, \quad A\begin{pmatrix}0 \\ 0 \\ 1\end{pmatrix} = \begin{pmatrix}1 \\ 1 \\ 2\end{pmatrix}.$ If $X = (x_1, x_2, x_3)^T$ and $I$ is an identity matrix of order $3$, then the system $(A - 2I)X = \begin{pmatrix}4 \\ 1 \\ 1\end{pmatrix}$ has:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 June Morning Shift) PYQ

Solution

Qus : 43 JEE MAIN PYQ

If for the matrix, $A = \left[ {\matrix{ 1 & { - \alpha } \cr \alpha & \beta \cr } } \right]$, $A{A^T} = {I_2}$, then the value of ${\alpha ^4} + {\beta ^4}$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Evening Shift) PYQ

Solution

Qus : 44 JEE MAIN PYQ

The following system of linear equations :- 2x + 3y + 2z = 9, 3x + 2y + 2z = 9, x $-$ y + 4z = 8

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Evening Shift) PYQ

Solution

Qus : 45 JEE MAIN PYQ

Let $A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]$. If M and N are two matrices given by $M = \sum\limits_{k = 1}^{10} {{A^{2k}}} $ and $N = \sum\limits_{k = 1}^{10} {{A^{2k - 1}}} $ then MN² is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 June Morning Shift) PYQ

Solution

Qus : 46 JEE MAIN PYQ

Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A² is 1, then the possible number of such matrices is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 February Morning Shift) PYQ

Solution

Qus : 47 JEE MAIN PYQ

Let $A$ and $B$ be two $3 \times 3$ non-zero real matrices such that $AB$ is a zero matrix. Then

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 July Morning Shift) PYQ

Solution

Qus : 48 JEE MAIN PYQ

$A=\left[\begin{array}{cc}1 & 5 \\ \lambda & 10\end{array}\right], \mathrm{A}^{-1}=\alpha \mathrm{A}+\beta \mathrm{I}$ and $\alpha+\beta=-2$, then $4 \alpha^{2}+\beta^{2}+\lambda^{2}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (8 April Evening Shift) PYQ

Solution

Qus : 49 JEE MAIN PYQ

Let $P=\begin{bmatrix}1&0&0\\[2pt]3&1&0\\[2pt]9&3&1\end{bmatrix}$ and $Q=[q_{ij}]$ be two $3\times 3$ matrices such that $Q-P^{5}=I_{3}$. Then $\displaystyle \frac{2q_{11}+q_{31}}{q_{32}}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Morning Shift) PYQ

Solution

Qus : 50 JEE MAIN PYQ

Let [$\lambda$] be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations x + y + z = 4, 3x + 2y + 5z = 3, 9x + 4y + (28 + [$\lambda$])z = [$\lambda$] has a solution is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 August Evening Shift) PYQ

Solution

Qus : 51 JEE MAIN PYQ

The system of equations

$ - kx + 3y - 14z = 25$

$ - 15x + 4y - kz = 3$

$ - 4x + y + 3z = 4$

is consistent for all k in the set

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 June Evening Shift) PYQ

Solution

Qus : 52 JEE MAIN PYQ

Let A = $\left[ {\matrix{ 1 & 0 & 0 \cr 1 & 1 & 0 \cr 1 & 1 & 1 \cr } } \right]$ and B = A²⁰. Then the sum of the elements of the first column of B is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (16 April Morning Shift) PYQ

Solution

Qus : 53 JEE MAIN PYQ

Let $A$ be a square matrix such that $AA^{\mathrm T}=I$. Then $\dfrac12\,A\Big[(A+A^{\mathrm T})^{2}+(A-A^{\mathrm T})^{2}\Big]$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Morning Shift) PYQ

Solution

Qus : 54 JEE MAIN PYQ

Consider the following system of equations : x + 2y $-$ 3z = a 2x + 6y $-$ 11z = bx $-$ 2y + 7z = c, where a, b and c are real constants. Then the system of equations :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 February Evening Shift) PYQ

Solution

Qus : 55 JEE MAIN PYQ

The system of linear equations $2x - y + 3z = 5$ $3x + 2y - z = 7$ $4x + 5y + \alpha z = \beta,$ which of the following is NOT correct?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (10 April Morning Shift) PYQ

Solution

Qus : 56 JEE MAIN PYQ

Which of the following matrices can NOT be obtained from the matrix $\begin{bmatrix}-1 & 2 \\ 1 & -1\end{bmatrix}$ by a single elementary row operation?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 July Evening Shift) PYQ

Solution

Qus : 57 JEE MAIN PYQ

The ordered pair (a, b), for which the system of linear equations

3x $-$ 2y + z = b

5x $-$ 8y + 9z = 3

2x + y + az = $-$1

has no solution, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 June Morning Shift) PYQ

Solution

Qus : 58 JEE MAIN PYQ

The set of all values of $\lambda$ for which the system of linear equations

$x-2y-2z=\lambda x$

$x+2y+z=\lambda y$

$-x-y=\lambda z$

has a non-trivial solution:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Evening Shift) PYQ

Solution

Qus : 59 JEE MAIN PYQ

If $A$ is a square matrix of order $3$ such that $\det(A) = 3$ and $\det(\text{adj}(-4,\text{adj}(-3,\text{adj}(3,\text{adj}((2A)^{-1}))))) = 2^m 3^n$, then $m + 2n$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Evening Shift) PYQ

Solution

Qus : 60 JEE MAIN PYQ

Let $A$ be a $2 \times 2$ real matrix with entries from $\{0,1\}$ and $|A|\neq 0$. Consider the following two statements: (P) If $A \neq I_2$, then $|A| = -1$ (Q) If $|A| = 1$, then $\operatorname{tr}(A) = 2$ where $I_2$ denotes $2 \times 2$ identity matrix and $\operatorname{tr}(A)$ denotes the sum of the diagonal entries of $A$.

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2 September 2020 (Morning) PYQ

Solution

Qus : 61 JEE MAIN PYQ

If the system of equations $x+y+z=6$ $2x+5y+\alpha z=\beta$ $x+2y+3z=14$ has infinitely many solutions, then $\alpha+\beta$ is equal to:

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4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 July Evening Shift) PYQ

Solution

Qus : 62 JEE MAIN PYQ

Let $A = \left[ {\matrix{ i & { - i} \cr { - i} & i \cr } } \right],i = \sqrt { - 1} $. Then, the system of linear equations ${A^8}\left[ {\matrix{ x \cr y \cr } } \right] = \left[ {\matrix{ 8 \cr {64} \cr } } \right]$ has :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (16 March Morning Shift) PYQ

Solution

Qus : 63 JEE MAIN PYQ

If the system of equations

$\alpha$x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = $\beta$

has infinitely many solutions, then the ordered pair ($\alpha$, $\beta$) is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 June Evening Shift) PYQ

Solution

Qus : 64 JEE MAIN PYQ

Let $\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]$ and $\mathrm{P}=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right], \theta>0$. If $\mathrm{B}=\mathrm{PAP}{ }^{\top}, \mathrm{C}=\mathrm{P}^{\top} \mathrm{B}^{10} \mathrm{P}$ and the sum of the diagonal elements of $C$ is $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$, then $m+n$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Evening Shift) PYQ

Solution

Qus : 65 JEE MAIN PYQ

If $A = \begin{bmatrix} 2 & -3 \\ -4 & 1 \end{bmatrix}$, then $\operatorname{adj}(3A^{2} + 12A)$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (Offline) PYQ

Solution

Qus : 66 JEE MAIN PYQ

Let $A=\begin{bmatrix} 2&1&2\\ 6&2&11\\ 3&3&2 \end{bmatrix} \quad\text{and}\quad P=\begin{bmatrix} 1&2&0\\ 5&0&2\\ 7&1&5 \end{bmatrix}. $ The sum of the prime factors of $\left|\,P^{-1}AP-2I\,\right|$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Evening Shift) PYQ

Solution

Qus : 67 JEE MAIN PYQ

Let $\mathbf{A}$ be a $2 \times 2$ matrix with real entries such that $\mathbf{A}' = \alpha \mathbf{A} + \mathbf{I}$, where $\alpha \in \mathbb{R} - \{-1, 1\}$. If $\det(\mathbf{A}^2 - \mathbf{A}) = 4$, then the sum of all possible values of $\alpha$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Morning Shift) PYQ

Solution

Qus : 68 JEE MAIN PYQ

Let $A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$. If $A^3=4 A^2-A-21 I$, where $I$ is the identity matrix of order $3 \times 3$, then $2 a+3 b$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (8 April Morning Shift) PYQ

Solution

Qus : 69 JEE MAIN PYQ

Let $A = \left( {\matrix{ {\cos \alpha } & { - \sin \alpha } \cr {\sin \alpha } & {\cos \alpha } \cr } } \right)$, ($\alpha $ $ \in $ R)
such that ${A^{32}} = \left( {\matrix{ 0 & { - 1} \cr 1 & 0 \cr } } \right)$ then a value of $\alpha $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Morning Shift) PYQ

Solution

Qus : 70 JEE MAIN PYQ

Let a, b, c $ \in $ R be all non-zero and satisfy a³ + b³ + c³ = 2. If the matrix A = $\left( {\matrix{ a & b & c \cr b & c & a \cr c & a & b \cr } } \right)$ satisfies A^TA = I, then a value of abc can be :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2 September 2020 (Evening) PYQ

Solution

Qus : 71 JEE MAIN PYQ

If $A$ and $B$ are two non-zero $n \times n$ matrices such that $A^{2}+B=A^{2}B$, then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (24 January Morning Shift) PYQ

Solution

Qus : 72 JEE MAIN PYQ

Let $ A = \begin{bmatrix} 2 & 2+p & 2+p+q \\ 4 & 6+2p & 8+3p+2q \\ 6 & 12+3p & 20+6p+3q \end{bmatrix} $.

If $ \det(\text{adj}(\text{adj}(3A))) = 2^m \cdot 3^n $, $ m, n \in \mathbb{N} $, then $ m + n $ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (8 April Evening Shift) PYQ

Solution

Qus : 73 JEE MAIN PYQ

Let f : (–1,$\infty $)$ \to $ R be defined by f(0) = 1 andLet A = {X = (x, y, z)^T: PX = 0 and

x² + y² + z² = 1} where

$P = \left[ {\matrix{ 1 & 2 & 1 \cr { - 2} & 3 & { - 4} \cr 1 & 9 & { - 1} \cr } } \right]$,

then the set A :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2 September 2020 (Evening) PYQ

Solution

Qus : 74 JEE MAIN PYQ

Let the system of linear equations $x + 2y + z = 2$, $\alpha x + 3y - z = \alpha $, $ - \alpha x + y + 2z = - \alpha $ be inconsistent. Then $\alpha$ is equal to :

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3
4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 June Evening Shift) PYQ

Solution

Qus : 75 JEE MAIN PYQ

Consider the system of linear equations $x + y + z = 4\mu,\quad x + 2y + 2\lambda z = 10\mu,\quad x + 3y + 4\lambda^2 z = \mu^2 + 15$ where $\lambda, \mu \in \mathbb{R}$. Which one of the following statements is NOT correct?

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4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Morning Shift) PYQ

Solution

Qus : 76 JEE MAIN PYQ

If the system of linear equations $7x + 11y + \alpha z = 13$ $5x + 4y + 7z = \beta$ $175x + 194y + 57z = 361$ has infinitely many solutions, then $\alpha + \beta + 2$ is equal to:

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4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Evening Shift) PYQ

Solution

Qus : 77 JEE MAIN PYQ

If the system of equations $x + 2y + 3z = 3$ $4x + 3y - 4z = 4$ $8x + 4y - \lambda z = 9 + \mu$ has infinitely many solutions, then the ordered pair $(\lambda,\mu)$ is equal to:

1
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3
4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (24 January Evening Shift) PYQ

Solution

Qus : 78 JEE MAIN PYQ

The number of real values of $\lambda$ for which the system of linear equations

$2x+4y-\lambda z=0$

$4x+\lambda y+2z=0$

$\lambda x+2y+2z=0$

has infinitely many solutions, is :

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3
4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

Qus : 79 JEE MAIN PYQ

Let α be a solution of $x^2 + x + 1 = 0$, and for some a and b in

$R, \begin{bmatrix} 4 & a & b \end{bmatrix} \begin{bmatrix} 1 & 16 & 13 \\ -1 & -1 & 2 \\ -2 & -14 & -8 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$. If $\frac{4}{\alpha^4} + \frac{m}{\alpha^a} + \frac{n}{\alpha^b} = 3$, then m + n is equal to _______

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (8 April Evening Shift) PYQ

Solution

Qus : 80 JEE MAIN PYQ

Let $A$ be any $3\times 3$ invertible matrix. Then which one of the following is not always true?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

Qus : 81 JEE MAIN PYQ

The number of square matrices of order $5$ with entries from the set $\{0,1\}$, such that the sum of all the elements in each row is $1$ and the sum of all the elements in each column is also $1$, is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (24 January Evening Shift) PYQ

Solution

Qus : 82 JEE MAIN PYQ

If \[ \begin{vmatrix} x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2 \end{vmatrix} = \dfrac{9}{8}\,(103x+81), \] then $\lambda,\ \dfrac{\lambda}{3}$ are the roots of the equation:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Evening Shift) PYQ

Solution

Qus : 83 JEE MAIN PYQ

The maximum value of $f(x) = \left| {\matrix{ {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\cos 2x} \cr {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr } } \right|,x \in R$ is :

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3
4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (16 March Evening Shift) PYQ

Solution

Qus : 84 JEE MAIN PYQ

Let $\mathrm{A}$ be a square matrix such that $\mathrm{AA}^{\mathrm{T}}=\mathrm{I}$. Then $\frac{1}{2} A\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right]$ is equal to

1
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3
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (29 January Morning Shift) PYQ

Solution

Qus : 85 JEE MAIN PYQ

Let $\mathbf{A}=\begin{bmatrix} 1 & \tfrac{1}{51} \\[2pt] 0 & 1 \end{bmatrix}$. If $\mathbf{B}=\begin{bmatrix} 1 & 2 \\ -1 & -1 \end{bmatrix}\mathbf{A}\begin{bmatrix} -1 & -2 \\ 1 & 1 \end{bmatrix}$, then the sum of all the elements of the matrix $\displaystyle \sum_{n=1}^{50} \mathbf{B}^n$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (12 April Morning Shift) PYQ

Solution

Qus : 86 JEE MAIN PYQ

Let $R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$ be a non-zero $3 \times 3$ matrix, where $x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0, \theta \in(0,2 \pi)$. For a square matrix $M$, let trace $(M)$ denote the sum of all the diagonal entries of $M$. Then, among the statements:

(I) Trace $(R)=0$

(II) If trace $(\operatorname{adj}(\operatorname{adj}(R))=0$, then $R$ has exactly one non-zero entry.

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Evening Shift) PYQ

Solution

Qus : 87 JEE MAIN PYQ

Let $S_{1}$ and $S_{2}$ be respectively the sets of all $a\in \mathbb{R}\setminus\{0\}$ for which the system of linear equations $ax+2ay-3az=1$ $(2a+1)x+(2a+3)y+(a+1)z=2$ $(3a+5)x+(a+5)y+(a+2)z=3$ has unique solution and infinitely many solutions. Then

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4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (25 January Morning Shift) PYQ

Solution

Qus : 88 JEE MAIN PYQ

Consider the system of linear equations $x+y+z=5,\quad x+2y+\lambda^2 z=9,\quad x+3y+\lambda z=\mu,$ where $\lambda,\mu\in\mathbb{R}$. Which of the following statements is NOT correct?

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4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Evening Shift) PYQ

Solution

Qus : 89 JEE MAIN PYQ

For the system of linear equations

$2 x+4 y+2 a z=b$

$x+2 y+3 z=4$

$2 x-5 y+2 z=8$

which of the following is NOT correct?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Morning Shift) PYQ

Solution

Qus : 90 JEE MAIN PYQ

If the function $f(x) = \left\{ {\matrix{ {(1 + |\cos x|)^{\lambda \over {|\cos x|}}} & , & {0 < x < {\pi \over 2}} \cr \mu & , & {x = {\pi \over 2}} \cr e^{{{\cot 6x} \over {{}\cot 4x}}} & , & {{\pi \over 2} < x < \pi } \cr } } \right.$

is continuous at $x = {\pi \over 2}$, then $9\lambda + 6{\log _e}\mu + {\mu ^6} - {e^{6\lambda }}$ is equal to

1
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3
4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (25 January Evening Shift) PYQ

Solution

Qus : 91 JEE MAIN PYQ

For two $3 \times 3$ matrices $A$ and $B$, let $A + B = 2B^T$ and $3A + 2B = I_3$, where $B^T$ is the transpose of $B$ and $I_3$ is $3 \times 3$ identity matrix. Then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

Qus : 92 JEE MAIN PYQ

Let $B=\left[\begin{array}{lll}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4\end{array}\right], \alpha > 2$ be the adjoint of a matrix $A$ and $|A|=2$. Then $\left[\begin{array}{ccc}\alpha & -2 \alpha & \alpha\end{array}\right] B\left[\begin{array}{c}\alpha \\ -2 \alpha \\ \alpha\end{array}\right]$$ is equal to :

1
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3
4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Morning Shift) PYQ

Solution

Qus : 93 JEE MAIN PYQ

If $\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix} \begin{bmatrix}1 & 2 \\ 0 & 1\end{bmatrix} \begin{bmatrix}1 & 3 \\ 0 & 1\end{bmatrix} \cdots \begin{bmatrix}1 & n-1 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}1 & 78 \\ 0 & 1\end{bmatrix}$, then the inverse of $\begin{bmatrix}1 & n \\ 0 & 1\end{bmatrix}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Morning Shift) PYQ

Solution

Qus : 94 JEE MAIN PYQ

The system of linear equations
3x - 2y - kz = 10
2x - 4y - 2z = 6
x+2y - z = 5m
is inconsistent if :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Morning Shift) PYQ

Solution

Qus : 95 JEE MAIN PYQ

Let $A=\begin{bmatrix}\dfrac{1}{\sqrt{10}} & \dfrac{3}{\sqrt{10}}\\[4pt]-\dfrac{3}{\sqrt{10}} & \dfrac{1}{\sqrt{10}}\end{bmatrix}$ and $B=\begin{bmatrix}1 & -i\\[2pt] 0 & 1\end{bmatrix}$, where $i=\sqrt{-1}$.

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (25 January Evening Shift) PYQ

Solution

Qus : 96 JEE MAIN PYQ

Let $\mathrm{A}=\left[\begin{array}{cc}\alpha & -1 \\ 6 & \beta\end{array}\right], \alpha>0$, such that $\operatorname{det}(\mathrm{A})=0$ and $\alpha+\beta=1$. If I denotes $2 \times 2$ identity matrix, then the matrix $(I+A)^8$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Morning Shift) PYQ

Solution

Qus : 97 JEE MAIN PYQ

$\text{The number of symmetric matrices of order }3\text{, with all entries from the set }{0,1,2,3,4,5,6,7,8,9}\text{ is:}$

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Morning Shift) PYQ

Solution

Qus : 98 JEE MAIN PYQ

Let $A + 2B = \left[ {\matrix{ 1 & 2 & 0 \cr 6 & { - 3} & 3 \cr { - 5} & 3 & 1 \cr } } \right]$ and $2A - B = \left[ {\matrix{ 2 & { - 1} & 5 \cr 2 & { - 1} & 6 \cr 0 & 1 & 2 \cr } } \right]$. If Tr(A) denotes the sum of all diagonal elements of the matrix A, then Tr(A) $-$ Tr(B) has value equal to

1
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (18 March Morning Shift) PYQ

Solution

Qus : 99 JEE MAIN PYQ

Let $\alpha$ and $\beta$ be real numbers. Consider a $3\times 3$ matrix $A$ such that $A^{2}=3A+\alpha I$. If $A^{4}=21A+\beta I$, then

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Morning Shift) PYQ

Solution

Qus : 100 JEE MAIN PYQ

If the system of linear equations

$\begin{aligned} & x-2 y+z=-4 \\ & 2 x+\alpha y+3 z=5 \\ & 3 x-y+\beta z=3 \end{aligned}$

has infinitely many solutions, then $12 \alpha+13 \beta$ is equal to

1
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4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Morning Shift) PYQ

Solution

Qus : 101 JEE MAIN PYQ

Let the system of linear equations4x + $\lambda$y + 2z = 0 ,2x $-$ y + z = 0 , $\mu$x + 2y + 3z = 0, $\lambda$, $\mu$$\in$R. has a non-trivial solution. Then which of the following is true?

1
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4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (18 March Evening Shift) PYQ

Solution

Qus : 102 JEE MAIN PYQ

If the following system of linear equations 2x + y + z = 5, x $-$ y + z = 3, x + y + az = b has no solution, then :

1
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4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (31 August Morning Shift) PYQ

Solution

Qus : 103 JEE MAIN PYQ

Consider the following system of equations \[ \begin{cases} \alpha x+2y+z=1,\\ 2\alpha x+3y+z=1,\\ 3x+\alpha y+2z=\beta \end{cases} \] for some $\alpha,\beta\in\mathbb{R}$. Then which of the following is NOT correct?

1
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4
Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Morning Shift) PYQ

Solution

Qus : 104 JEE MAIN PYQ

The system of linear equations

$ x + \lambda y - z = 0 $

$ \lambda x - y - z = 0 $

$ x + y - \lambda z = 0 $

has a non-trivial solution for:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (Offline) PYQ

Solution

Qus : 105 JEE MAIN PYQ

Let $A$ be a $3\times3$ real matrix such that \[ A\!\begin{pmatrix}1\\0\\1\end{pmatrix} =2\!\begin{pmatrix}1\\0\\1\end{pmatrix},\qquad A\!\begin{pmatrix}-1\\0\\1\end{pmatrix} =4\!\begin{pmatrix}-1\\0\\1\end{pmatrix},\qquad A\!\begin{pmatrix}0\\1\\0\end{pmatrix} =2\!\begin{pmatrix}0\\1\\0\end{pmatrix}. \] Then, the system $(A-3I)\!\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}1\\2\\3\end{pmatrix}$ has:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Evening Shift) PYQ

Solution

Qus : 106 JEE MAIN PYQ

If ${a_r} = \cos {{2r\pi } \over 9} + i\sin {{2r\pi } \over 9}$, r = 1, 2, 3, ....., i = $\sqrt { - 1} $, then the determinant $\left| {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{a_4}} & {{a_5}} & {{a_6}} \cr {{a_7}} & {{a_8}} & {{a_9}} \cr } } \right|$ is equal to :<

1
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4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (31 August Morning Shift) PYQ

Solution

Qus : 107 JEE MAIN PYQ

$A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A \,\text{adj}\, A = A\,A^{T}$, then $5a + b$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (Offline) PYQ

Solution

Qus : 108 JEE MAIN PYQ

If $\alpha$ + $\beta$ + $\gamma$ = 2$\pi$, then the system of equations :- x + (cos $\gamma$)y + (cos $beta$)z = 0,(cos $\gamma$)x + y + (cos $\alpha$)z = 0(cos $\beta$)x + (cos $\alpha$)y + z = 0 has :

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4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (31 August Evening Shift) PYQ

Solution

Qus : 109 JEE MAIN PYQ

If the system of linear equations

2x + y $-$ z = 7

x $-$ 3y + 2z = 1

x + 4y + $\delta$z = k, where $\delta$, k $\in$ R has infinitely many solutions, then $\delta$ + k is equal to:

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4
Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 June Morning Shift) PYQ

Solution

Qus : 110 JEE MAIN PYQ

Let $A$ be a $3 \times 3$ real matrix such that $A^2(A-2 I)-4(A-I)=O$, where $I$ and $O$ are the identity and null matrices, respectively. If $A^5=\alpha A^2+\beta A+\gamma I$, where $\alpha, \beta$, and $\gamma$ are real constants, then $\alpha+\beta+\gamma$ is equal to :

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4
Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Evening Shift) PYQ

Solution

Qus : 111 JEE MAIN PYQ

Let $A = [{a_{ij}}]$ be a square matrix of order 3 such that ${a_{ij}} = {2^{j - i}}$, for all i, j = 1, 2, 3. Then, the matrix A² + A³ + ...... + A¹⁰ is equal to :

1
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4
Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 June Morning Shift) PYQ

Solution

Qus : 112 JEE MAIN PYQ

Let $B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$ and $A$ be a $2 \times 2$ matrix such that $A B^{-1}=A^{-1}$. If $B C B^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$, then $2 \beta-\alpha$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (9 April Evening Shift) PYQ

Solution

Qus : 113 JEE MAIN PYQ

If $f(x)=[x]-\left[\dfrac{x}{4}\right],\ x\in\mathbb{R}$, where $[\cdot]$ denotes the greatest integer function, then

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Evening Shift) PYQ

Solution

Qus : 114 JEE MAIN PYQ

The system of linear equations

$x + y + z = 2$

$2x + 3y + 2z = 5$

$2x + 3y + (a^2 - 1)z = a + 1$

then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Morning Shift) PYQ

Solution

Qus : 115 JEE MAIN PYQ

Let $A = \left[ {\matrix{ 2 & 3 \cr a & 0 \cr } } \right]$, a$\in$R be written as P + Q where P is a symmetric matrix and Q is skew symmetric matrix. If det(Q) = 9, then the modulus of the sum of all possible values of determinant of P is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (20 July Morning Shift) PYQ

Solution

Qus : 116 JEE MAIN PYQ

Let $A = [{a_{ij}}]$ be a 3 $\times$ 3 matrix, where ${a_{ij}} = \left\{ {\matrix{ 1 & , & {if\,i = j} \cr { - x} & , & {if\,\left| {i - j} \right| = 1} \cr {2x + 1} & , & {otherwise.} \cr } } \right.$ Let a function f : R $\to$ R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (20 July Morning Shift) PYQ

Solution

Qus : 117 JEE MAIN PYQ

If $P = \begin{bmatrix} \dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \\ -\dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \end{bmatrix}$ and $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, $Q = P A P^{T}$, then $P^{T} Q^{2015} P$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (9 April Morning Shift) PYQ

Solution

Qus : 118 JEE MAIN PYQ

If $A = \left[ {\matrix{ {\cos \theta } & {i\sin \theta } \cr {i\sin \theta } & {\cos \theta } \cr } } \right]$, $\left( {\theta = {\pi \over {24}}} \right)$

and ${A^5} = \left[ {\matrix{ a & b \cr c & d \cr } } \right]$, where $i = \sqrt { - 1} $ then which one of the following isnot true?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Morning) PYQ

Solution

Qus : 119 JEE MAIN PYQ

For $x \in \mathbb{R} - \{0,1\}$, let $f_1(x)=\dfrac{1}{x}$, $f_2(x)=1-x$, and $f_3(x)=\dfrac{1}{1-x}$ be three given functions. If a function $J(x)$ satisfies $(f_2 \circ J \circ f_1)(x)=f_3(x)$, then $J(x)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Morning Shift) PYQ

Solution

Qus : 120 JEE MAIN PYQ

Let the system of linear equations \[ \begin{cases} x + y + kz = 2,\\ 2x + 3y - z = 1,\\ 3x + 4y + 2z = k \end{cases} \] have infinitely many solutions. Then the system \[ \begin{cases} (k+1)x + (2k-1)y = 7,\\ (2k+1)x + (k+5)y = 10 \end{cases} \] has:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (30 January Morning Shift) PYQ

Solution

Qus : 121 JEE MAIN PYQ

Consider the system of linear equations$-$x + y + 2z = 0 3x $-$ ay + 5z = 12x, $-$ 2y $-$ az = 7, Let S₁ be the set of all a$\in$R for which the system is inconsistent and S₂ be the set of all a$\in$R for which the system has infinitely many solutions. If n(S₁) and n(S₂) denote the number of elements in S₁ and S₂ respectively, then

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (1 September Evening Shift) PYQ

Solution

Qus : 122 JEE MAIN PYQ

If $A=\begin{bmatrix}\sqrt2&1\\-1&\sqrt2\end{bmatrix}$, $B=\begin{bmatrix}1&0\\1&1\end{bmatrix}$, $C=ABA^{\mathrm T}$ and $X=A^{\mathrm T}C^{2}A$, then $\det X$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Morning Shift) PYQ

Solution

Qus : 123 JEE MAIN PYQ

Let ${J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx} $, $\forall$ n > m and n, m $\in$$ N. Consider a matrix $A = {[{a_{ij}}]_{3 \times 3}}$ where $${a_{ij}} = \left\{ {\matrix{ {{j_{6 + i,3}} - {j_{i + 3,3}},} & {i \le j} \cr {0,} & {i > j} \cr } } \right.$. Then $\left| {adj{A^{ - 1}}} \right|$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (1 September Evening Shift) PYQ

Solution

Qus : 124 JEE MAIN PYQ

If the system of equations
x+y+z=2
2x+4y–z=6
3x+2y+$\lambda $z=$\mu $
has infinitely many solutions, then

1
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4
Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Evening) PYQ

Solution

Qus : 125 JEE MAIN PYQ

For $\alpha,\beta\in\mathbb{R}$, suppose the system of linear equations $\begin{aligned} x-y+z&=5,\\ 2x+2y+\alpha z&=8,\\ 3x-y+4z&=\beta \end{aligned}$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (30 January Evening Shift) PYQ

Solution

Qus : 126 JEE MAIN PYQ

If the system of linear equations

$x+y+z=5$

$x+2y+2z=6$

$x+3y+\lambda z=\mu,; (\lambda,\mu\in\mathbb{R})$

has infinitely many solutions, then the value of $\lambda+\mu$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 April Morning Shift) PYQ

Solution

Qus : 127 JEE MAIN PYQ

Let y = y(x) satisfies the equation ${{dy} \over {dx}} - |A| = 0$, for all x > 0, where $A = \left[ {\matrix{ y & {\sin x} & 1 \cr 0 & { - 1} & 1 \cr 2 & 0 & {{1 \over x}} \cr } } \right]$. If $y(\pi ) = \pi + 2$, then the value of $y\left( {{\pi \over 2}} \right)$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (20 July Evening Shift) PYQ

Solution

Qus : 128 JEE MAIN PYQ

Suppose the vectors x₁, x₂ and x₃ are the solutions of the system of linear equations, Ax = b when the vector b on the right side is equal to b₁, b₂ and b₃ respectively. if${x_1} = \left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right]$, ${x_2} = \left[ {\matrix{ 0 \cr 2 \cr 1 \cr } } \right]$, ${x_3} = \left[ {\matrix{ 0 \cr 0 \cr 1 \cr } } \right]$${b_1} = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]$, ${b_2} = \left[ {\matrix{ 0 \cr 2 \cr 0 \cr } } \right]$ and ${b_3} = \left[ {\matrix{ 0 \cr 0 \cr 2 \cr } } \right]$, then the determinant of A is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Evening) PYQ

Solution

Qus : 129 JEE MAIN PYQ

The value of k $\in$R, for which the following system of linear equations. 3x $-$ y + 4z = 3,x + 2y $-$ 3z = $-$2, 6x + 5y + kz = $-$3,has infinitely many solutions, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (20 July Evening Shift) PYQ

Solution

Qus : 130 JEE MAIN PYQ

If $A = \begin{bmatrix} -4 & -1 \\ 3 & 1 \end{bmatrix}$, then the determinant of the matrix $(A^{2016} - 2A^{2015} - A^{2014})$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (10 April Morning Shift) PYQ

Solution

Qus : 131 JEE MAIN PYQ

Let $A$ be a $3 \times 3$ matrix such that $A^{2} - 5A + 7I = 0$. \textbf{Statement I:} $A^{-1} = \dfrac{1}{7}(5I - A)$. \textbf{Statement II:} The polynomial $A^{3} - 2A^{2} - 3A + I$ can be reduced to $5(A - 4I)$. Then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (10 April Morning Shift) PYQ

Solution

Qus : 132 JEE MAIN PYQ

If $A=\begin{bmatrix} e^{t} & e^{-t}\cos t & e^{-t}\sin t\\[4pt] e^{t} & -e^{-t}\cos t - e^{-t}\sin t & -e^{-t}\sin t + e^{-t}\cos t\\[4pt] e^{t} & 2e^{-t}\sin t & -2e^{-t}\cos t \end{bmatrix}$, then $A$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Evening Shift) PYQ

Solution

Qus : 133 JEE MAIN PYQ

The number of $\theta \in(0,4 \pi)$ for which the system of linear equations

$\begin{aligned}&3(\sin 3 \theta) x-y+z=2 \\\\&3(\cos 2 \theta) x+4 y+3 z=3 \\\\&6 x+7 y+7 z=9\end{aligned}$

has no solution, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 July Morning Shift) PYQ

Solution

Qus : 134 JEE MAIN PYQ

The values of $\lambda$ and $\mu$ such that the system of equations $x + y + z = 6$, $3x + 5y + 5z = 26$, $x + 2y + \lambda z = \mu $ has no solution, are :

1
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4
Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (22 July Evening Shift) PYQ

Solution

Qus : 135 JEE MAIN PYQ

Let A = [a_ij] be a real matrix of order 3 $\times$ 3, such that a_i1 + a_i2 + a_i3 = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix A³ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (22 July Evening Shift) PYQ

Solution

Qus : 136 JEE MAIN PYQ

Let the system of equations $x+2y+3z=5,\quad 2x+3y+z=9,\quad 4x+3y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda+2\mu$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Evening Shift) PYQ

Solution

Qus : 137 JEE MAIN PYQ

$ \textbf{Q:}$ For the system of linear equations $x + y + z = 6,\ \alpha x + \beta y + 7z = 3,\ x + 2y + 3z = 14$, which of the following is **NOT true**?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (31 January Morning Shift) PYQ

Solution

Qus : 138 JEE MAIN PYQ

If the system of linear equations :

$\begin{aligned} & x+y+2 z=6 \\ & 2 x+3 y+\mathrm{az}=\mathrm{a}+1 \\ & -x-3 y+\mathrm{b} z=2 \mathrm{~b} \end{aligned}$

where $a, b \in \mathbf{R}$, has infinitely many solutions, then $7 a+3 b$ is equal to :

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4
Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (22 January Evening Shift) PYQ

Solution

Qus : 139 JEE MAIN PYQ

If the system of linear equations

$x-4y+7z=g$,

$3y-5z=h$,

$-2x+5y-9z=k$

is consistent, then:

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4
Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Evening Shift) PYQ

Solution

Qus : 140 JEE MAIN PYQ

Let $A=\begin{pmatrix}1&0&0\\[2pt]0&4&-1\\[2pt]0&12&-3\end{pmatrix}$. Then the sum of the diagonal elements of the matrix $(A+I)^{11}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (31 January Morning Shift) PYQ

Solution

Qus : 141 JEE MAIN PYQ

For a $3\times3$ matrix $M$, let $\operatorname{trace}(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a $3\times3$ matrix such that $|A|=\dfrac{1}{2}$ and $\operatorname{trace}(A)=3$. If $B=\operatorname{adj}(\operatorname{adj}(2A))$, then the value of $|B|+\operatorname{trace}(B)$ equals:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (22 January Evening Shift) PYQ

Solution

Qus : 142 JEE MAIN PYQ

The number of real values of $\lambda$, such that the system of linear equations

$2x - 3y + 5z = 9$

$x + 3y - z = -18$

$3x - y + (\lambda^2 - |\lambda|)z = 16$

has no solutions, is

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Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 July Evening Shift) PYQ

Solution

Qus : 143 JEE MAIN PYQ

If the system of equations

$x + y + z = 5$

$x + 2y + 3z = 9$

$x + 3y + az = \beta$

has infinitely many solutions, then $\beta - \alpha =$

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4
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 January Morning Shift) PYQ

Solution

Qus : 144 JEE MAIN PYQ

Let $\alpha \in (0,\infty)$ and $A=\begin{bmatrix}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{bmatrix}$. If $\det(\operatorname{adj}(2A-A^T)\cdot\operatorname{adj}(A-2A^T))=2^8$, then $(\det(A))^2$ is equal to:

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Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Morning Shift) PYQ

Solution

Qus : 145 JEE MAIN PYQ

The values of a and b, for which the system of equations 2x + 3y + 6z = 8, x + 2y + az = 5, 3x + 5y + 9z = b, has no solution, are :

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3
4
Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Morning Shift) PYQ

Solution

Qus : 146 JEE MAIN PYQ

If $B = \left[ {\matrix{ 5 & {2\alpha } & 1 \cr 0 & 2 & 1 \cr \alpha & 3 & { - 1} \cr } } \right]$ is the inverse of a 3 × 3 matrix A, then the sum of all values of $\alpha $ for which det(A) + 1 = 0, is :

1
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4
Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Morning Shift) PYQ

Solution

Qus : 147 JEE MAIN PYQ

If $A=\begin{bmatrix} 1 & 2 & 2\\ 2 & 1 & -2\\ a & 2 & b \end{bmatrix}$ is a matrix satisfying the equation $AA^{T}=9I$, where $I$ is $3\times 3$ identity matrix, then the ordered pair $(a,b)$ is equal to :

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Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2015 (Offline) PYQ

Solution

Qus : 148 JEE MAIN PYQ

If the system of linear equations
x + y + 3z = 0
x + 3y + k²z = 0
3x + y + 3z = 0
has a non-zero solution (x, y, z) for some k $ \in $ R,then x + $\left( {{y \over z}} \right)$ is equal to :

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Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 5 September 2020 (Evening) PYQ

Solution

Qus : 149 JEE MAIN PYQ

If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = $\left[ {\matrix{ 2 & 3 \cr 5 & { - 1} \cr } } \right]$, then AB is equal to :

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Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Morning Shift) PYQ

Solution

Qus : 150 JEE MAIN PYQ

If the system of equations $(\lambda-1)x+(\lambda-4)y+\lambda z=5$ $\lambda x+(\lambda-1)y+(\lambda-4)z=7$ $(\lambda+1)x+(\lambda+2)y-(\lambda+2)z=9$ has infinitely many solutions, then $\lambda^2+\lambda$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (23 January Morning Shift) PYQ

Solution

JEE MAIN

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